The key component in the [1]} framework is a family of noise operators \(\lbrace M_z\rbrace \) , referred to as Noise Operators with Leakage. Formally, given a string \(z \in \lbrace \bot ,\top \rbrace ^t\)  the corresponding noise operator \(M_z\)  on the space \(\Omega ^t\)  is defined as follows. For any \(\omega \in \Omega ^t\) , one can sample \(\omega ^{\prime } \sim M_z(\omega )\)  using the following process. For every \(i \in [t]\) , do the following independently: if \(z(i) = \top \) , set \(\omega ^{\prime }(i) = \omega (i)\)  otherwise re-sample \(\omega ^{\prime }(i) \sim \Omega \) . In particular, for a random draw of \(\lbrace \bot ,\top \rbrace ^t_\beta \) , the noise operator \(M_z\)  behaves like the standard noise operator \(T_\beta \)  on the space \(L_2(\Omega )\) . By incorporating the above noise operator in the averaging step, one can guarantee that the spectral gap of the averaging operator is at least \(1 - \beta \) , thus guaranteeing the aforementioned concentration on the biases. We describe our overall reduction in Figure REF .
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